Question

Solve the differential equation \(x^2 \frac{∂u}{∂x}+y^2 \frac{∂u}{∂y}=u \) using the method of separation of variables if \(u(0,y) = e^{\frac{2}{y}} \).

(a) \(e^{\frac{-3}{y}} e^{\frac{2}{x}} \)

(b) \(e^{\frac{3}{y}} e^{\frac{2}{x}} \)

(c) \(e^{\frac{-3}{x}} e^{\frac{2}{y}} \)

(d) \(e^{\frac{3}{x}} e^{\frac{2}{y}} \)

I have been asked this question during an internship interview.

My doubt is from Solution of PDE by Variable Separation Method topic in division Applications of Partial Differential Equations of Engineering Mathematics

Answer 1

The correct choice is (c) \(e^{\frac{-3}{x}} e^{\frac{2}{y}} \)

Best explanation: \(u(x,t) = X(x) T(t) \)

\(x^2 X’Y+y^2 Y’X=XY\)

\(X = ce^{\frac{-k}{x}} \)

\(Y = c’e^{\frac{k-1}{y}} \)

\(u(x,t) = cc’ e^{\frac{-k}{x}} e^{\frac{k-1}{y}} \)

\(u(0,y) = e^{\frac{2}{y}}= cc’ e^{\frac{k-1}{y}} \)

k = 3 and cc’ = 1

Therefore \(u(x,t) = e^{\frac{-3}{x}} e^{\frac{2}{y}}. \)